Gruber P. Convex and Discrete Geometry

Hence
�
S d−1
6 Mixed Volumes and Quermassintegrals 107
v(F|u ⊥ �
) dσ(u) = v(F) |u F · u| dσ(u) = 2κd−1v(F).
S(P) = �
F facet of P
= 1
�
κd−1
Sd−1 v(F) = 1
S d−1
�
2κd−1
Sd−1 v(P|u ⊥ ) dσ(u).
� �
F facet of P
v(F|u ⊥ ) � dσ(u)
Second, let C be a proper convex body. We may assume that o ∈ int C. Bythe
proof of Theorem 7.4, there is a sequence (Pn) of proper convex polytopes such that
�
Pn ⊆ C ⊆ 1 + 1
�
Pn and Pn → C as n →∞.
n
S(·) = dW1(·) is continuous by Theorem 6.13 (iii). Thus
(2) S(Pn) → S(C) as n →∞.
The functions v(Pn|u ⊥ ) : u ∈ S d−1 are continuous in u for each n = 1, 2,... Since
v(Pn|u ⊥ ) ≤ v(C|u ⊥ �
) ≤ 1 + 1
�d−1 v(Pn|u
n
⊥ ) for u ∈ S d−1 ,
and since v(C|u⊥ ) is bounded on Sd−1 , the function v(C|u⊥ ) : u ∈ Sd−1 is the
uniform limit of the continuous functions v(Pn|u ⊥ ). Thus it is continuous itself.
Integration over Sd−1 then shows that
�
v(Pn|u ⊥ �
) dσ(u) ≤ v(C|u ⊥ �
) dσ(u) ≤ 1 + 1
� �
d−1
v(Pn|u
n
⊥ ) dσ(u).
S d−1
S d−1
Since, by the first part of the proof,
we conclude that
S(Pn) = 1
S(Pn) → 1
�
κd−1
Sd−1 �
κd−1
Sd−1 Together with (2), this shows that
S(C) = 1
κd−1
Sd−1 v(Pn|u ⊥ ) dσ(u),
S d−1
v(C|u ⊥ ) dσ(u) as n →∞.
�
v(C|u ⊥ ) dσ(u).